A number is called a $y$-rough number, if it has no prime divisor below $y$.
The number of rough numbers in an interval, lets say, $[10^{99},10^{100}]$ is approximately the length of the interval multiplied with $\prod1-\frac{1}{p}$, where $p$ runs over the primes below $y$.
However, if $\ 2\cdot3\cdot5\cdot...\cdot p$ is not much smaller than the length of the interval, the approximation should have a considerable deviation of the actual value.
The article in Wikipedia only gives asymptotic formulas, so my questions :
How accurate is the naive approximation ?
How can I determine the number of $y$-rough numbers in an interval reasonably accurate ?
This is related to Buchstab function.
Let $\Phi(x,y)$ be the number of positive integers $n\leq x$ which do not have any prime divisor $p< y$. Then it is known that for any fixed $U>1$
$$ \Phi(x,y)=\frac{x w(u)}{\log y} - \frac y{\log y} +O\left(\frac x{(\log x)^2} \right) $$ where $y=x^{\frac 1u}$ with $1\leq u\leq U$ , $y\geq 2$, and $w(u)$ satisfies $$ w(u)=\frac 1u \ \ \textrm{ for }1\leq u \leq 2, $$ $$ (uw(u))'=w(u-1) \ \ \textrm{ for }u>2. $$ Also, it is known that $w(u)\rightarrow e^{-\gamma}$ as $u\rightarrow\infty$.
You may find more information in Montgomery & Vaughan's 'Multiplicative Number Theory', Classical Theory, Chapter 7.